Switched-capacitor filters provide a practical method for the fully-integrated realization of high-quality filters. The basic motivation for the development of the switched-capacitor filter was the need to obtain fully-integrated high-quality frequency-selective devices. Integrated circuits cannot contain high-Q inductors, since the quality factor Q decreases with decreasing size. Resistors can be integrated, but they are nonlinear, occupy a large area on the chip, and their tolerances and temperature coefficients do not track with those of the capacitors on the same chip. Hence, neither the time constants nor the pole positions of the filter can be accurately controlled. High-quality capacitors, by contrast, can be realized conveniently in an MOS integrated circuit. The dielectric material is SiO.sub.2, an excellent insulator. The electrodes can be made of metal, or polycrystalline silicon, or heavily doped crystalline silicon. In any presently available construction, the structure is planar, positioned parallel with the surface of the chip. Hence the bottom plate, which is in the substrate or close to it, is coupled to the substrate by a stray capacitance. The stray capacitance is usually 5-20% of the main capacitance, and is nonlinear. Due to the interconnecting lines connected to the top plate of the main capacitor, a stray capacitance also exists between the top plate and ground (substrate). It is typically 0.1-5% of the main capacitance and also may be nonlinear.
Good switches can also be fabricated using MOS technology. The on-resistance depends on the area allowed for the MOS transistor used as the switch; 1000 ohms can readily be attained. The off-resistance is, for practical purposes, infinite. The stray capacitors between the gate (where a large clock signal exists, turning the switch on or off) and drain and source (connected to the rest of the switched-capacitor filter) are of the order of 2-10 fF. They can play an important role in causing clock-signal feedthrough and dc offset.
A key component of switched-capacitor filters is the active element, almost always an operational amplifier (opamp). Opamps with up to 90 dB dc gain, unity-gain bandwidth of 5 MHz and dc power drain of 2 mW are now practical. In terms of chip area, an opamp occupies about as much space as a 50 pF capacitor. It is also (along with clock feedthrough) the major source of noise in the switched-capacitor filter.
Because of the availability and properties of the components described above, the switched-capacitor filter utilizes only capacitors, switches and opamps. Superficially, a capacitor with one or more associated switches can be used to first store and then dissipate electric energy, and thus act as a simulated resistor. Using capacitors, simulated resistors and opamps, one can also simulate inductors and/or FDNR's. Hence, in principle, any passive and/or active filter response can be obtained by a switched-capacitor filter. From dimensional considerations, it can readily be seen that the voltage gain of such a circuit will depend only on the ratios of capacitances, which can be closely controlled.
The basic concept behind the switched-capacitor filter is known in the prior art and is illustrated in FIG. 1. The concept is based on resistance simulation. Referring to FIG. 1 and assuming that the variation of the voltages v.sub.1 and v.sub.2 during a clock period T is negligible, the charge q(t) which enters the input terminal and leaves the output terminal between t and t+T is given by [v.sub.1 (t)-v.sub.2 (t)]C. Thus, the average current is [v.sub.1 (t)-v.sub.2 (t)]/(T/C), the same as would flow through a resistor R=T/C connected between the input and output terminals. Thus, for signals whose highest frequency component is much less than the clock frequency f.sub.c =1/T, either of the circuits of FIG. 1 will simulate such a resistor. This analogy has opened up the whole wealth of active filter realizations for conversion to equivalent switched-capacitor filter circuits. The basic building block in all these switched-capacitor filters is the equivalent of the active -RC integrator (FIG. 2). The transfer function of the circuit is ##EQU1## where s is the Laplace-transform variable. Replacing R by either of the equivalents in FIG. 1, the SC form is obtained.
Unfortunately, this simple approach gives good results only if a very large ratio (say, 50 or more) of the clock frequency to the highest signal component can be maintained. To see the limitations involved, consider using the circuit of FIG. 1b to simulate R in FIG. 2 (FIG. 3). At t=tn nT (where .nu. is an integer), C.sub.1 acquires a charge C.sub.1 v.sub.in (t.sub.n); at t=t.sub.n 1/2 nT+T/2, it dumps the charge into C.sub.2. Thus, v.sub.out satisfies ##EQU2##
It should be carefully noted that the foregoing equation is valid only because v.sub.in (t) is assumed to be a sampled-and-held waveform--otherwise, v.sub.in (t.sub.n +T) would have to be used (FIG. 3c). Also, it follows from FIGS. 3a and b that v.sub.out (t) is also a staircase signal, so that v.sub.out (t.sub.n+1/2)=v.sub.out (t.sub.n+1). Thus, regarding v.sub.out (t.sub.n) and v.sub.in (t.sub.n) as sequences, and taking the z-transform of (2), ##EQU3## results. The desired frequency response, from (1), is H(j)=-1/(j RC); the one obtained from (3) is ##EQU4## If 0&lt;.omega.T&lt;&lt;1, then .vertline.cos .omega.T-1.vertline.&lt;&lt;sin .omega.T.perspectiveto..omega.T, so that H'.perspectiveto.-(C.sub.1 /C.sub.2)/j.omega.T. Thus, for C.sub.2 /C.sub.1 =RC/T, H'(e.sup.j.omega.T).perspectiveto.H (j.omega.).
Using the circuit of FIG. 3, or other similar integrators, switched-capacitor filters can be constructed successfully. However, for the important case of narrowband bandpass filters, the resulting circuit is too sensitive to variations of the capacitance values and the imperfections of the amplifiers. Hence, only other principles, such as that of the pseudo-N-path filter, are practical.
An understanding of an N-path filter is helpful to the understanding of a switched-capacitor pseudo-N-path filter. N-path filters were introduced originally for analog filter realization. The basic block diagram for a 3-path filter (N=3) is shown in FIG. 4. In this system, a number of unwanted "mirror frequencies" are generated along with the desired signal. If a perfect match exists between the N paths, then the phasors of the unwanted mirror frequencies form a polygon with zero resultant. Otherwise they appear at the output, including a component at the center of the passband.
Pseudo-N-path analog filters were developed to overcome this sensitivity to path mismatch. In these filters only one physical path exists; however, each memory-possessing element in the path is connected to a circulating delay line, which discharges and recharges these elements such that the overall circuit represents various paths in different clock phases. This approach thus eliminated the passband distortion due to mismatch of filter's paths; however, the circuit was complicated and the imperfections of the delay devices affected the performance.
The N-path principle was then extended to switched-capacitor filters. N=4 was used and a clock frequency of N.omega.c was applied to the switched-capacitor filter. The circuit was obtained by the LDI s-to-z transformation of an analog prototype; thus, some loss distortion was introduced due to the approximation used for terminations (although this is not very significant for narrow-band bandpass filters). All memory possessing capacitors were replaced by N identical capacitors which were commutated to provide the N signal paths. Obviously, any mismatch between these capacitors resulted in path mismatch, and introduced unwanted frequencies in the passband.
It was subsequently found that by using the frequency transformation z.fwdarw.z .sup.N, and the bilinear s-to-z transformation, an exact low-path to N-path mapping could be obtained. The resulting circuit was free from some approximation errors inherent in the earlier circuits. The z.fwdarw.z.sup.N transformation corresponds to an N.sup.th -order reactance transformation of the analog prototype filter. A straightforward application of this procedure, however, resulted in a circuit which needed more memory-possessing capacitors than that of the earlier circuits, and since each of these capacitors should be replaced by N identical capacitors, the circuit becomes more complicated and thus more vulnerable to mismatch of paths.
Therefore, while the prior art teaches the concept of the pseudo-N-path switched capacitor filter, the resulting circuits are very cumbersome, and are susceptible to the path mismatch problem, with the resulting unwanted frequencies in the output. Furthermore, the previous circuits have the undesirable requirement of either using a recirculating analog shift register or requiring multiplexed capacitors. Finally, the cumbersome nature of the prior art circuits makes them generaly unsuitable for monolithic integration.
Accordingly, it is the principal object of the present invention to reduce and eliminate distortion in a pseudo-N-path switched-capacitor filter.
It is a further object of the present invention to reduce multiplexing of capacitors in a switched-capacitor pseudo-N-path filter.
It is yet another object of the present invention to produce a switched-capacitor pseudo-N-path filter with a minimum number of components.